分形值随机迭代算法与分形族
A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy
摘要：
我们描述了新的家庭的随机分形，被称为“V-变量”，这是中间的概念之间的确定性和标准的随机分形。参数V描述了“可变性”的程度：在每个放大水平下，任何V-变量分形最多有V个关键的“形式”或“形状”。V-变量随机分形具有令人惊讶的性质，它们可以使用前向过程进行计算。更确切地说，通常的随机迭代算法的一个版本，对集合（或测量）而不是点进行操作，可以用于对每个族进行采样。为了提出这一理论，我们回顾分形（分形措施），确定性和随机的相关结果。然后，我们的新的结果是通过构造一个迭代函数系统（一个超级IFS）从一组标准IFS以及相应的一组概率。超IFS的吸引子被称为超分形;它是V-变量随机分形（集合或测度）的集合，以及在这个集合上的相关概率分布。当底层空间例如是$\mathbb{R}^{2}$，并且变换在计算上是直接的（诸如仿射变换）时，可以借助于在存储器使用方面非常高效的算法来对超分形进行采样。通过算例说明了该算法。一些变种，特殊情况下，概括的框架，和潜在的应用。
英文摘要：
We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability" : at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example $\mathbb{R}^{2}$, and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.
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